peak and superparamagnetic effects in
S. H. Mahmood1, I. Bsoula2
1Physics Department, The University of Jordan, Amman, Jordan. Phone: (962) 796709673
2Physics Department, Al al-Bayt University, Mafraq 130040, Jordan. Phone: (962 2) 6297000 (ext.
3588), Fax: (962 2) 6297031
Abstract. In this article, the thermomagnetic properties of a system of Ga-substituted
barium hexaferrite nanoparticles (BaFe12-xGaxO19) prepared by ball milling were
investigated. The thermomagnetic curves for the samples with x ranging from 0.0 to 1.0
exhibited sharp peaks with high magnetization just below TC (Hopkinson peaks). The
height of the peak for our samples was similar or larger than previously observed or
calculated values. Theoretical treatment of the experimental data demonstrated that the
peaks are due to the effect of superparamagnetic relaxations of the magnetic particle.
This effect was confirmed by hysteresis measurements at, and just below the
temperature at which the peak occurred. Consequently, the particle diameters were
calculated from the experimental data using a theoretical model based on the
superparamagnetic behavior of a system of uniaxial, randomly oriented, single domain,
non-interacting particles. The calculated diameters of 11 - 26 nm are less than the
physical diameters determined from TEM measurements. The factors responsible for the
low calculated values are discussed.
Barium hexaferrite BaFe12O19 (BaM) possess interesting properties such as large saturation
magnetization, high coercivity, high Curie temperature, large uniaxial magnetic anisotropy and
chemical stability. These materials have been investigated due to their importance for both
fundamental research and technological applications in permanent magnets, high-density magnetic
recording, magneto-optics and microwave devices [1-6]. Different techniques have been used to
prepare and characterize hexaferrite particles [7-13]. The magnetic properties of these materials have
been tuned by substitution of Fe by different magnetic and nonmagnetic cations, and intensively
investigated by different techniques [14-20].
The magnetization of certain hexaferrites exhibits a peak (Hopkinson peak) near TC in the
thermomagnetic curve in a weak applied magnetic field. Popov and Mikhov explained this effect
using Stoner-Wohlfarth model for magnetically stable, single-domain (SD), randomly oriented
particles . According to this model, the magnetization is given by:
a e-mail : Ibrahimbsoul@yahoo.com
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29 00039 (2012)
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where p is the packing fraction of the powder, Ms(T) is the bulk saturation magnetization, and Ha(T)
is the anisotropy field at temperature T. The explanation was based on the argument that the
competition between the increase in magnetization due to the decrease in the anisotropy field, and
the decrease in magnetization due to the decrease in saturation magnetization as the temperature
increases may result in a peak in the magnetization. Later, the effect of the demagnetizing field was
taken into consideration in explaining the origin of Hopkinson peak . Using a completely
different approach, the origin of the peak was explained by the superparamagnetic behavior of
magnetic particles which do not exhibit a peak as a result of the variations of the saturation
magnetization and anisotropy field of the material .
In the present work we prepared a system of BaM nanoparticles doped with Gallium, and
investigated its structural and magnetic properties. The Hopkinson peak height was analyzed in
terms of the superparamagnetic relaxation processes of the particles and compared with the
experimental data to arrive at a conclusion concerning the mechanism responsible for the peak, and
the particle size distribution of the synthesized powders.
2 Experimental procedures
BaFe12-xGaxO19 powders with x ranging from 0.0 to 1.0 were prepared using high energy ball
milling and appropriate heat treatment. The samples were characterized using XRD and transmission
electron microscopy (TEM), and the magnetic measurements were performed using a vibrating
sample magnetometer (VSM). For further details on the experimental procedures the reader is
referred to our earlier publication .
3 Results and discussion
Fig. 1. Standard JCPDS pattern for M-type hexagonal barium ferrite (file no: 043-0002) and XRD
patterns of BaFe12-xGaxO19 with different doping concentration.
Fig. 1 shows the XRD patterns of samples of BaFe12-xGaxO19 along with the standard pattern
(JCPDS: 043-0002) for hexagonal barium ferrite (BaFe12O19) with space group P63/mmc. No
secondary phases were detected in the diffraction patterns indicating the formation of a pure phase
with variations in the lattice parameters less than 0.1%.
The average crystallite size was determined using Scherrer formula ,
where D is the crystallite size, k the Scherrer constant (= 0.94), λ the wavelength of radiation
(1.54056 Å), β the peak width at half maximum measured in radians, and θ the peak position. The
average crystallite size for the pure and doped samples ranges from 37 nm to 45 nm.
Fig. 2. TEM images of BaFe12-xGaxO19, a) x = 0.0, b) x =1.0
TEM images of representative samples are shown in Fig. 2. The average particle size for the pure
sample is (42 ± 13) nm, and for the sample with x = 1.0 is (41 ± 13) nm. These values indicate that
the synthesized powders consist of single domain magnetic nanoparticles with a relatively narrow
particle size distribution.
Fig. 3. Saturation magnetization and coercivity variations with x for BaFe12-xGaxO19
Table 1: Coercivity, Saturation magnetization, remanence ratio (Mrs = Mr/Ms), and Curie temperature for Ga-
substituted hexaferrite samples
x Hc (kOe)
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The initial magnetization of the pure sample was checked and found to increase slowly at low
fields followed by a rapid increase at higher fields . This behavior is typical for randomly
oriented single domain magnetic particles. The variations of the saturation magnetization and
coercivity with x are shown in Fig. 3, and their values together with Curie temperatures for the
samples are listed in Table 1. The behavior of the saturation magnetization and coercivity indicates
that Ga ions replace Fe ions at both spin-up and spin-down sites. The small initial drop in coercivity
and slow decrease in saturation magnetization for x values up to 0.2 are consistent with the
substitution of Ga at spin-up 2a and spin-down 4f1 sites with preference for occupying 2a sites. This
is consistent with the substitution of small amounts of Ti-Ru at these sites as confirmed by
Mossbauer spectroscopy . For higher x values, spin-up 12k sites which contribute negatively to
the anisotropy field start getting occupied by Ga ions, leading to the observed increase in coercivity
and decrease in saturation magnetization. The change in behavior of Ms at x = 0.6 suggests that
beyond this value, the fraction of Ga ions substituting Fe ions at spin-down sites remains constant at
a value of 0.2, where the remaining fraction substitute Fe ions at spin-up 12k sites. This substitution
would lead to the observed 5% drop in Ms at x = 0.6, and the 15% drop at x = 1.0. The remanence
ratio Mrs = Mr/Ms is ~ 0.5 (Table 1), which is consistent with the theoretical value for a system of
uniaxial, single domain, randomly oriented particles.
Fig. 4. Thermomagnetic curves of BaFe12-xGaxO19.
Fig. 5. Experimental and calculated thermomagnetic curves of bulk barium hexaferrite.
Fig.4 shows the thermomagnetic curves as a function of temperature for the samples at a constant
applied field of 100 Oe. All curves exhibit sharp pronounced Hopkinson peaks just below TC, and the
height of the peak relative to the minimum magnetization (RPH) for all samples is shown in Table 2.
The sharpness of the peaks indicates a narrow particle size distribution. The relative peak heights for
our samples are similar or higher than the observed value of about 10 for Co-Ti substituted sample,
and the calculated value of about 8 based on superparamagnetic relaxation . To investigate the
origin of the peak, a sample is prepared from bulk Barium hexaferrite (Aldrich made) powder of
grain size ~ 0.5 - 2 µm (with small fraction of smaller particles), and Ms for the sample is measured
against temperature. The anisotropy field Ha for the sample is determined from the switching field
distribution evaluated by differentiating the reduced DC demagnetization curve . The
magnetization is then calculated from these values by adopting Stoner-Wohlfarth model for SD
blocked particles (eq. (1)) and the results are shown in Fig. 5. The figure shows only a very small
rise in the magnetization below TC which is insignificant compared with the peak heights observed
for our samples. Further, the magnetization is measured versus T for the bulk sample (Fig. 5). The
measured magnetization shows a relatively small sharp peak with relative height of 1.8, and a
behavior similar to that of the calculated magnetization in the temperature range below the peak. The
higher values of the measured magnetization in this temperature range could be associated with the
effects of interparticle interactions, or the presence of multidomain particles in the sample, which are
not accounted for in Stoner-Wohlfarth theory. Thus the observed sharp peaks in the thermomagnetic
measurements on our samples cannot be due to the temperature dependences of the saturation
magnetization and anisotropy field. The small peak observed for the bulk sample could be associated
with a small fraction of superparamagnetic particles in the bulk powder, or with the temperature
dependence of the saturation magnetization and the anisotropy field [21, 22]. However, the relative
height of this peak cannot account for the large observed peaks in our samples. Accordingly, we are
led to believe that the observed Hopkinson peaks in our synthesized samples are associated with the
superparamagnetic relaxations of the particles in the samples.
Fig. 6. Hysteresis loops for the sample with x = 0 at different temperatures.
The magnetic relaxations of the particles are further confirmed by measuring the hysteresis loops
at the peak temperature and at lower temperatures. Fig. 6 shows the loops for the sample with x = 0,
which show superparamagnetic behavior with almost zero coercivity at the peak temperature, and the
appearance of coercivity at the temperature of the minimum magnetization just before the rise of the
peak (at 460 ºC). The increase in coercivity as the temperature is lowered is a consequence of the
gradual blocking of the particles as the temperature is lowered. All samples show similar behavior,
as illustrated in Fig. 7 for the sample with x = 0.4.
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Fig. 7. Hysteresis loops for the sample with x = 0.4 at different temperatures.
Assuming that the volumes of the superparamagnetic particles have a flat top distribution
between V1 and V2, the upper and lower limits of the particle volumes can be calculated following
the model calculation in . Following this model, the upper limit is calculated from the initial
susceptibility given by:
where Ms is the saturation magnetization at temperature T. The lower limit is calculated from the
slope of the magnetization versus 1/H in the high field region, which is given by :
Fig. 8. Magnetization curve for the sample with x = 0.4 at peak temperature (414 oC).
The initial susceptibility, saturation magnetization, and the slope of M vs. 1/H are determined
from the magnetization curve (Fig. 8) for each sample at the peak temperature where the sample
Table 2: Hopkinson peak temperature (Tp), saturation magnetization at the peak temperature, particle
diameters, and relative peak height (RPH) for Ga-substituted hexaferrite samples.
Ms(Tp)(emu/g) D1(nm) D2(nm) RPH
453 5.83 12 14 10.8
428 7.12 12 13 9.3
414 10.07 12 22 8.9
398 9.72 13 26 12
382 8.12 12 20 10.5
362 8.72 11 14 10.8
The calculated particle diameters, assuming spherical particles, are listed in Table 2. The particle
diameters range from 11 nm – 26 nm, which are lower than previously reported results .
Differences between particle diameters evaluated from the magnetic data and the previously reported
values could be due to deviation of the particle size distribution from the assumed flat top
distribution, and to interparticle interactions. However, these effects are possibly not enough to
account for the observed reduction of more than 50%. A number of reasons, in addition to the
assumptions of the theory, could be responsible for the low calculated values. Firstly, the calculated
volume is the volume of the magnetic core of the particle, and a nonmagnetic shell (dead layer)
could be surrounding the particles, which would give smaller particle sizes than the physical sizes.
Secondly, the particles could be platelets in shape rather than spherical as suggested by the TEM
images. A rough estimate of the platelet thickness could be obtained by assuming that the observed
physical diameter is that of a cylindrical platelet of volume equals to that calculated from the
magnetic data. In this case, the thickness of the platelets is found to be between 1.5 nm and 4.0 nm.
The lower limit of the thickness is smaller than the lattice parameter c, which is evidence that the
magnetic volume is an under estimate of the physical volume.
Barium hexaferrite nanoparticle systems doped with different concentrations of Ga prepared by ball
milling exhibit single hexagonal phase with crystallite size ranging between 37 and 45 nm. The
magnetic measurements as a function of temperature exhibit sharp peaks with high relative
magnetization which cannot be explained on the basis of Stoner-Wohlfarth model for an assembly of
randomly oriented, non-interacting, single-domain particles. The magnetization curves at the peak
temperatures for all samples are consistent with the behavior of a system of superparamagnetic
particles, which are blocked at slightly lower temperature, indicating narrow superparamagnetic
particle size distribution. The calculated particle diameters for these samples are between 11 nm and
26 nm. These values suggest that the samples consist of single magnetic domain particles.
This work is supported by a generous grant from the Scientific Research Support Fund (SRF) in
Jordan under grant number (S/1/21/2009). The work is accomplished during a sabbatical leave
provided by Yarmouk University to one of the authors (S.H.M.) which was spent at Al al-Bayt
University. The Authors would like to thank Munir Khdour, Yarmouk University, for his technical
assistance in electron microscopy.
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